Generalization of Abel's solution for both magnetoionic components in the real-height problem

摘要

Modem methods of calculating the electron density distribution from virtual heights include joint use of both magnetoionic components since, in certain ranges, one component alone does not provide sufficient information to obtain a unique solution of the integral equation involved. Experience shows, however, that joint use of both components does not always give satisfactory results. We therefore consider in this paper whether sufficient information is provided by two components, used together, to remove ambiguities in the problem. A solution formula is first developed for each component, in a form similar to the Abel (1823) solution valid for an isotropic ionosphere. Application of these generalized solution formulas to each component and a combination of the two resulting expressions leads to a new integral equation in which only the distribution of the underlying ionization or the distribution in the valley between two layers is involved. Consideration of this new integral equation leads to the result that in general only a first- or second-order approximation to the distribution in those parts can be obtained.